Finding maximum coplanar points
There is no "simple" way to do anything in numerical analysis. However, the tasks can be laid out straightforwardly.

EDIT: This idea is a brain-fart because it doesn't distinguish parallel planes. The Internet must never forget though. Pile on, people.

How about this: generate an array of normalized normal vectors (not vectors in the dynamic array sense; in the math sense. A 1x3 array representation of a floating-point ordered triple is fine. I guess you can make an object but why bother) for every 3-point combination (not permutation) of the n points you have in your set. This list will contain nC3 normal "vectors".

If no plane that can be made contains more than 3 points in the set, then your list of normal vectors will entirely contain nC3 unique normals.

If any plane exists with 4 vertices, you will have 4C3 = 4 identical normal vectors in that list.

If any plane exists with 5 vertices, you will have 5C3 = 10 identical normals.

If any plane exists with k vertices, where 3 &lt; k &lt; n, you will have kC3 identical normals.

You would just need to quantify the normals that are the same, identify the largest value, and deduce what k was.
EDIT: Thanks to Rich Sedman for pointing out comparison should be done with dot-product. Weeding out parallel planes is another task that would need to be done optimally in the worst case scenario (a stack of parallel planes).

Doing all this would require knowing how to do a 3D cross product, how to normalize a vector, how to find a dot-product, and how to solve this equation:

k!/(3!(k-3)!) = count, Or:

k(k-1)(k-2)(k-3)!/(3!(k-3)!) = count, Or:

k(k-1)(k-2)/6 = count,

where count is the number of duplicated normal vectors,  which you will have already counted before solving this. This is just a 3rd degree polynomial.

Note also that if the set is randomly chosen, and even if the range and precision of floating point values are small, it is very unlikely that you will have more than 3 points in any plane. The list would need to be curated to have more than 3 coplanar points in order to make good use of this work.